Some analogous results for ring of operators on Hilbert space

Abstract

In this research paper we explained and see that the results for ring of operators on Hilbert space is same as the ring, whenever we suppose H be a Hilbert space and R be a set of operators on H given by R={Ti:Ti2=O, TiTj=O for all i,j} then (R,+) is an additive abelian group and (R,.) is multiplicative group as well as the algebraic structure (R, +,.) forms a ring. Also a special case achieved that If (R,+,.) is a Boolean ring then it is commutative ring. Also we got a beautiful result as (R,+,.) is commutative if and only if (Ti + Tj)2 = Ti2 + 2TiTj + Tj2. We got result on homomorphism and isomorphism also as Let R and R1 be rings of operators on a Hilbert space H and f: R→R1 be a mapping defined by f(Ti)=Ti, then f is Homomorphism as well as Isomorphism. Thus here we have seen some important results of the algebraic structure Ring is the same of the Ring of operators on a Hilbert Space.